The automorphisms of endomorphism semigroups of relatively free groups

Abstract

The question of describing the automorphisms of [Formula: see text] for a free algebra [Formula: see text] in a certain variety was considered by different authors since 2002. In this paper, we consider this question for the class of all relatively free groups having only cyclic centralizers of non-trivial elements. We prove that each automorphism of the endomorphism semigroup [Formula: see text] of groups [Formula: see text] from this class is uniquely determined by its action on the subgroup of inner automorphisms [Formula: see text]. The obtained general result includes the following cases: absolutely free groups, free Burnside groups of odd period [Formula: see text], free groups of infinitely based varieties of Adian (the cardinality of the set of such varieties is continuum), and so on.